TITLE: What are two continuous maps from $S^1$ to $S^1$ which are not homotopic?
QUESTION [11 upvotes]: This is an exam question I encountered while studying for my exam for our topology course:

Give two continuous maps from $S^1$ to $S^1$ which are not homotopic. (Of course, provide a proof as well.)

The only continuous maps from $S^1$ to $S^1$ I can think of are rotations, and I thought rotations on a circle can be continuously morphed into one another.

REPLY [4 votes]: $F_1:S^1\to S^1 $
$F_1(s)=(\cos(2\pi s),\sin(2\pi s))$ and
$F_2:S^1\to S^1$ 
$F_2(s)=(\cos(4\pi s),\sin(4\pi s))$ where $s$ goes from $0$ to $1$, are not homotopic.